Theorem 9 MAX CUT is -inapproximable for all , under a different complexity assumption, UGC.

Raghavendra 2009.

Theorem 10 Let MAX BLAH be any constraint satisfaction problem. There is an SDP-based algorithm which -approximates MAX BLAH. And there is a matching inapproximability result: MAX BLAH is -inapproximable for all , assuming UGC.

Pisier: can this be phrased in terms of a ratio ? Answer: this would be weaker.

1.2. How can one prove such inapproximability results ?

These are NP-hardness results. They are proved by Karp-reduction from known NP-hard problems.

The fact that -COVER is NP-hard can be stated: MAX -COVER is -inapproximable assuming PNP. To prove this, one proves that -COLORABILITY reduces in polynomial time -approximating MAX -COVER. This means producing a polytime algorithm that, given a graph , outputs sets and satisfying

Completeness. -colorable there exist sets covering -fraction of .

Soundness. not -colorable for all -tuples of sets, they cover -fraction of .

Proof: Introduce a -cover gadget for each vertex and edge pair. Attach them to the given graph, getting a set . take number of vertices. Completeness will hold by design. Soundness is proved by contrapositive. If there were sets covering -fraction of , then one can decode it into a -colouring of .

More details to be found in standard textbooks.

The strategy for Feige’s -inapproximability theorem goes along similar lines, except that the Soundness result is stronger. The reduction is longer, it goes through intermediate problems like LABEL COVER (also called MAX PROJECTION). In fact, many hardness results I will discuss start from hardness of -approximating MAX PROJECTION. This relies on the PCP theorem and Parallel Repetition.

Definition 11 For all integers , the MAX PROJECTION -problem has

– as input a bipartite graph , , regular, and for each edge , a projection constraint, i.e. a map ;

– as output assignments , .

– the goal is to maximize the fraction of consistent edges, i.e. .

One can think as and as label or color sets, and as coloring rules which depend on each edge.

The following is the heart of Feige’s inapproximability theorem.

Theorem 12 For all , there exist , such that MAX PROJECTION is -inapproximable assuming PNP.

Périfel: Can one prove inapproximability ? Answer: It is in fact stronger (not so obvious, but easy).